# Linear OT mapping estimation

```# Author: Remi Flamary <remi.flamary@unice.fr>
#

# sphinx_gallery_thumbnail_number = 2
```
```import os
from pathlib import Path

import numpy as np
from matplotlib import pyplot as plt
import ot
```

## Generate data

```n = 1000
d = 2
sigma = .1

rng = np.random.RandomState(42)

# source samples
angles = rng.rand(n, 1) * 2 * np.pi
xs = np.concatenate((np.sin(angles), np.cos(angles)),
axis=1) + sigma * rng.randn(n, 2)
xs[:n // 2, 1] += 2

# target samples
anglet = rng.rand(n, 1) * 2 * np.pi
xt = np.concatenate((np.sin(anglet), np.cos(anglet)),
axis=1) + sigma * rng.randn(n, 2)
xt[:n // 2, 1] += 2

A = np.array([[1.5, .7], [.7, 1.5]])
b = np.array([[4, 2]])
xt = xt.dot(A) + b
```

## Plot data

```plt.figure(1, (5, 5))
plt.plot(xs[:, 0], xs[:, 1], '+')
plt.plot(xt[:, 0], xt[:, 1], 'o')
plt.legend(('Source', 'Target'))
plt.title('Source and target distributions')
plt.show()
```

## Estimate linear mapping and transport

```# Gaussian (linear) Monge mapping estimation
Ae, be = ot.gaussian.empirical_bures_wasserstein_mapping(xs, xt)

xst = xs.dot(Ae) + be

# Gaussian (linear) GW mapping estimation
Agw, bgw = ot.gaussian.empirical_gaussian_gromov_wasserstein_mapping(xs, xt)

xstgw = xs.dot(Agw) + bgw
```

## Plot transported samples

```plt.figure(2, (10, 5))
plt.clf()
plt.subplot(1, 2, 1)
plt.plot(xs[:, 0], xs[:, 1], '+')
plt.plot(xt[:, 0], xt[:, 1], 'o')
plt.plot(xst[:, 0], xst[:, 1], '+')
plt.legend(('Source', 'Target', 'Transp. Monge'), loc=0)
plt.title('Transported samples with Monge')
plt.subplot(1, 2, 2)
plt.plot(xs[:, 0], xs[:, 1], '+')
plt.plot(xt[:, 0], xt[:, 1], 'o')
plt.plot(xstgw[:, 0], xstgw[:, 1], '+')
plt.legend(('Source', 'Target', 'Transp. GW'), loc=0)
plt.title('Transported samples with Gaussian GW')
plt.show()
```

## Load image data

```def im2mat(img):
"""Converts and image to matrix (one pixel per line)"""
return img.reshape((img.shape[0] * img.shape[1], img.shape[2]))

def mat2im(X, shape):
"""Converts back a matrix to an image"""
return X.reshape(shape)

def minmax(img):
return np.clip(img, 0, 1)

this_file = os.path.realpath('__file__')
data_path = os.path.join(Path(this_file).parent.parent.parent, 'data')

I1 = plt.imread(os.path.join(data_path, 'ocean_day.jpg')).astype(np.float64) / 256
I2 = plt.imread(os.path.join(data_path, 'ocean_sunset.jpg')).astype(np.float64) / 256

X1 = im2mat(I1)
X2 = im2mat(I2)
```

## Estimate mapping and adapt

```# Monge mapping
mapping = ot.da.LinearTransport()
mapping.fit(Xs=X1, Xt=X2)

xst = mapping.transform(Xs=X1)
xts = mapping.inverse_transform(Xt=X2)

I1t = minmax(mat2im(xst, I1.shape))
I2t = minmax(mat2im(xts, I2.shape))

# gaussian GW mapping

mapping = ot.da.LinearGWTransport()
mapping.fit(Xs=X1, Xt=X2)

xstgw = mapping.transform(Xs=X1)
xtsgw = mapping.inverse_transform(Xt=X2)

I1tgw = minmax(mat2im(xstgw, I1.shape))
I2tgw = minmax(mat2im(xtsgw, I2.shape))
```

## Plot transformed images

```plt.figure(3, figsize=(14, 7))

plt.subplot(2, 3, 1)
plt.imshow(I1)
plt.axis('off')
plt.title('Im. 1')

plt.subplot(2, 3, 4)
plt.imshow(I2)
plt.axis('off')
plt.title('Im. 2')

plt.subplot(2, 3, 2)
plt.imshow(I1t)
plt.axis('off')
plt.title('Monge mapping Im. 1')

plt.subplot(2, 3, 5)
plt.imshow(I2t)
plt.axis('off')
plt.title('Inverse Monge mapping Im. 2')

plt.subplot(2, 3, 3)
plt.imshow(I1tgw)
plt.axis('off')
plt.title('Gaussian GW mapping Im. 1')

plt.subplot(2, 3, 6)
plt.imshow(I2tgw)
plt.axis('off')
plt.title('Inverse Gaussian GW mapping Im. 2')
```
```Text(0.5, 1.0, 'Inverse Gaussian GW mapping Im. 2')
```

Total running time of the script: (0 minutes 1.320 seconds)

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